On the \(L^p\)-estmates for the weak solution of divergence form elliptic equations (Q2706025)

The author of this interesting paper investigates the regularity of the derivatives of the solution for an equation that belongs to a certain class of linear elliptic equations in divergence form, NEWLINE\[NEWLINE-\partial /\partial x_j(a_{ij}\partial /\partial x_i)u + b_iu_{x_i}-(d_iu)_{x_i}+cu=(f_j)_{x_j}.NEWLINE\]NEWLINE Here the author considers the Dirichlet problem in an open bounded \(n\geq 3\) dimensional domain with \(C^{1,1}\) boundary. The coefficients \(a_{ij}\) belong to the subspace of the John-Nirenberg class whose elements have norm on the balls vanishing as the radius of the balls approaches zero. The main result being proposed here includes the existence, uniqueness and regularity obtained by the technique based on perturbations of fundamental solutions of certain equations with constant coefficients. Some interesting explicit representation formulas in terms of singular integral operators are obtained and discussed as well.